How can I make my particles emit from the emitter object moving by key animation, instead of the emitter's default position?
Particle System has a bake option you wil see in the Panel.  You will probably need it.

If you have a previous bake.  Delete the bake.  Then bake again.

If you never baked.  Try baking for the first time.

Remember to bake after every significant change.  The particle system will not let you change parameters when a bake is active.  Yet you can change other objects such as physics obstacles. I have not checked every version of Blender 2.8X yet at one point there were obvious defects in the Particle System.